Question 44
Let $ABCD$ be a quadrilateral such that $\angle ABC = \angle CDA = 90^{\circ}$, and $BC = 7$. Let $E$ and $F$ be on $BD$ such that $AE$ and $CF$ are perpendicular to $BD$. Suppose that $BE = 3$. Determine the product of the smallest and largest possible lengths of $DF$.
